Ken'ichi OHSHIKA

I have been studying 3-manifolds and discrete groups. Although 3-manifold topology has a long tradition of research, which started with the pioneering work of Poincaré back in the 19th century, it is still one of the most active fields in topology. In the 1980's, Thurston published a famous conjecture called the geometrisation conjecture, stating that all compact 3-manifolds would be decomposed canonically into geometric pieces each of which has a locally homogeneous metric. Recently Perelman claimed that he has succeeded in solving this conjecture. If his claim is true, then the research of 3-manifolds is reduced to that of hyperbolic ones, which have metrics of constant sectional curvature -1. I am studying hyperbolic 3-manifolds from the viewpoint of Kleinian groups which have been an important topic in complex analysis. Kleinian groups are typical examples of discrete groups in Lie groups. More generally, it is in vogue to study groups as geometric objects regarding them as discrete groups by endowing them with the word metric, and I am also interested in this field. In particular, such things as hyperbolic groups invented by Gromov or isometric group actions on R-trees are closely related to the study of Kleinian groups. More general objects called automatic groups, whose operations are governed by automata, are also important objects in geometric group theory. Although geometric group theory is a relatively new field, it is promised to flourish in the near future.